Further Remarks on the Exponent of Convergence and the Hausdorff Dimension of the Limit Set of Kleinian Groups

Abstract

In [Patterson, Further remarks on the exponent of convergence of Poincaré series, Tôhoku Math. Journ. 35 (1983), 357–373], it was shown how to construct for a given ε > 0 a Kleinian group of the first kind with exponent of convergence smaller than ε. We show the more general result that for any $m \in \mathbb{N}$ there are Kleinian groups acting on (m + 1)-dimensional hyperbolic space for which the Hausdorff dimension of their uniformly radial limit set is less than a given arbitrary number d ∈ (0, m) and the Hausdorff dimension of their Jørgensen limit set is equal to a given arbitrary number j ∈ [0, m). Additionally, our result clarifies which part of the limit set gives rise to the result of Patterson’s original construction.The key idea in our construction is to combine the previous techniques of Patterson with a description of various limit sets in terms of the coding map.